Graham's Law of DiffusionFoundation

Concept

At the same temperature all gases share the same average kinetic energy, so lighter molecules move faster: diffusion/effusion rate ∝ 1/√M. Ammonia (17) beats HCl (36.5) down a glass tube — the white NH₄Cl ring forms closer to the HCl end, in exactly the √ ratio.

Key formula

Derivation

Equal average KE: $\tfrac12 M_A\overline{v_A^2} = \tfrac12 M_B\overline{v_B^2}$ ⇒ $v_A/v_B = \sqrt{M_B/M_A}$.

Rate of escape through a pinhole (effusion) ∝ molecular speed, giving Graham's law. Meeting point: distances ∝ rates, so the ring sits at $\frac{r_A}{r_A+r_B}$ of the tube from A.

Scenarios to explore

• Graham's Law — Diffusion race — rate ∝ 1/√M, NH₃ vs HCl tube.

Real-world applications

• Uranium enrichment: ²³⁵UF₆ effuses 1.0043× faster than ²³⁸UF₆ — thousands of cascade stages.
• Helium leak detection.
• Why H₂S smell travels slower than NH₃ sting.

JEE exam tips

• NH₃/HCl tube: ring at √(36.5/17) : 1 ≈ 1.47 : 1 → about 59% from the NH₃ end.
• Rates can be volumes per time: r = V/t → V₁t₂/V₂t₁ = √(M₂/M₁).
• Density form works because ρ ∝ M at same T,P.

Common mistakes

• Rate ratio with masses NOT square-rooted.
• Inverting: the LIGHTER gas is faster (M in the denominator under the root).
• Diffusion vs effusion sloppiness (effusion = through a pinhole; Graham strictly governs effusion).

Exam traps to avoid

• Equal volumes effusing: TIME ratio is the inverse of the rate ratio.
• Mixture problems: each gas effuses at its own rate — the escaping mixture enriches in the lighter gas.